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Question

Figure shows a uniform disc of radius R, from which a hole of radius R/2 has been cut out from left of the centre and is placed on the right of the centre of the disc. Find the COM of the resulting disc.



A
(R2,0)
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B
(R4,0)
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C
(R2,R2)
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D
(R4,R4)
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Solution

The correct option is B (R4,0)
Since the mass is distributed uniformally about x-axis, it can be said that y coordinate of centre of mass of the resulting disc will lie at the origin.
ycom=0
Now to find the x-coordinate of centre of mass we need to consider each part of the resulting disc separately.

Thus taking the disc of diameter R and mass m1 separately. It is evident that centre of mass of this disc will lie at its centre which is at a distance R/2 from the origin.
Now the centre of mass of the original disc of mass M without cutting the disc was at origin i.e (0,0) and since the mass was equally distributed, the cut out disc which is 1/4 th part of original disc (in terms of area) will have mass(m1) as M/4.

thus for original disc,
0=m1(R2)+m2(x)M(1)
Here m1=M/4 and m2=MM4=3M4
Substituting these in (1), we get
x=R/6
This is the centre of mass of the left out part of mass m2.

Now the x-coordinate of centre of mass of resulting disc (xcom) will be,
xcom=m2(R6)+m1(R2)M
xcom=3M4(R6)+M4(R2)M
xcom=R4

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